# An explicit self-dual construction of complete cotorsion pairs in the relative context

### Leonid Positselski

Czech Academy of Sciences, Prague, Czechia

## Abstract

Let $R→A$ be a homomorphism of associative rings, and let $(F,C)$ be a hereditary complete cotorsion pair in $R−Mod$. Let $(F_{A},C_{A})$ be the cotorsion pair in $A−Mod$ in which $F_{A}$ is the class of all left $A$-modules whose underlying $R$-modules belong to $F$. Assuming that the $F$-resolution dimension of every left $R$-module is finite and the class $F$ is preserved by the coinduction functor $Hom_{R}(A,−)$, we show that $C_{A}$ is the class of all direct summands of left $A$-modules finitely (co)filtered by $A$-modules coinduced from $R$-modules from $C$. Assuming that the class $F$ is closed under countable products and preserved by the functor $Hom_{R}(A,−)$, we prove that $C_{A}$ is the class of all direct summands of left $A$-modules cofiltered by $A$-modules coinduced from $R$-modules from $C$, with the decreasing filtration indexed by the natural numbers. A combined result, based on the assumption that countable products of modules from $F$ have finite $F$-resolution dimension bounded by $k$, involves cofiltrations indexed by the ordinal $ω+k$. The dual results also hold, provable by the same technique going back to the author's monograph on semi-infinite homological algebra~(2010). In addition, we discuss the $n$-cotilting and $n$-tilting cotorsion pairs, for which we obtain better results using a suitable version of a classical Bongartz–Ringel lemma. As an illustration of the main results of the paper, we consider certain cotorsion pairs related to the contraderived and coderived categories of curved DG-modules.

## Cite this article

Leonid Positselski, An explicit self-dual construction of complete cotorsion pairs in the relative context. Rend. Sem. Mat. Univ. Padova 149 (2023), pp. 191–253

DOI 10.4171/RSMUP/118